I and j as subscript and superscript?

[Physicsphysics]

I've come across countless equations in which i and j are written as either subscript or superscript or both. I am an AS level student (16 - 17 yrs old), and all I know is that i, j and k are unit vectors, but I've never seen them written as subscript or superscript and I'd like to understand what that notation means and how it differs from what I've seen in class.

This is the equation I am currently trying to decipher:

dl² = g_ij dxⁱ dx^j

Which is in this document:

http://www2.warwick.ac.uk/fac/sci/physics/current/teach/module_home/px436/notes/lecture20.pdf

Thanks in advance ^-^

 

[jedishrfu]

This has to do with contravariant and covariant notations which expresses how vectors and tensors transform from one coordinate system to another.

Wikipedia has some write-ups on them:

https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

To make sense of these concepts, you have to think in terms of spherical or cylindrical coordinates and not simply in xyz cartesian coordinates.

The units of measure is a good although simple example where the transformation involves axes changing units from meters to kilometers the drawn vector length of a velocity vector is inversely proportional (contra-variant) as it is shown smaller aka 10m/s vs 0.01km/sec.

Extra credit: For your extra credit homework tonight, write up an insight article on these two concepts and have it on my desk by 8am tomorrow. :-)

 

[micromass]

To make sense of these concepts, you have to think in terms of spherical or cylindrical coordinates and not in xyz cartesian coordinates.

I'm sorry, I don't follow. What does Einstein notation have to do with spherical or cylindrical coordinates? And why doesn't it work in cartesian? This is a very strange remark, I think.

 

[jedishrfu]

I'm sorry, I don't follow. What does Einstein notation have to do with spherical or cylindrical coordinates? And why doesn't it work in cartesian? This is a very strange remark, I think.

I was adding that to my post. I wanted to make the point of unit vectors along tangent lines vs unit vectors from normals to tangent planes not lining up as shown in the wiki diagrams. In simple xyz transformations you can't see the reason for covariance or contravariance easily.

Hopefully @micromass can clarify this better.

 

[micromass]

I was adding that to my post. I wanted to make the point of unit vectors along tangent lines vs unit vectors from normals to tangent planes not lining up as shown in the wiki diagrams. In simple xyz transformations you can't see the reason for covariance or contravariance easily.

Got it.

In either case. The OP should first study linear algebra, and particularly the notion of dual spaces and multilinear maps (tensors). Then he should study the notion of differential forms and basic differential geometry. That were the prerequisites I needed to grasp these subtle stuff.

 

[DrGreg]

I've come across countless equations in which i and j are written as either subscript or superscript or both. I am an AS level student (16 - 17 yrs old), and all I know is that i, j and k are unit vectors, but I've never seen them written as subscript or superscript and I'd like to understand what that notation means and how it differs from what I've seen in class.

This is the equation I am currently trying to decipher:

dl² = g_ij dxⁱ dx^j

Which is in this document:

http://www2.warwick.ac.uk/fac/sci/physics/current/teach/module_home/px436/notes/lecture20.pdf

Thanks in advance ^-^

The use of $i$ and $j$ in this context has no connection with the letters $\textbf{i}$, $\textbf{j}$, $\textbf{k}$ used to denote unit vectors. It's just coincidence that the same letters are being used with different meanings.
$$g_{ij} \, dx^i \, dx^j$$
is a shorthand notation which really means
$$ \sum_{i=1}^3 \sum_{j=1}^3 g_{ij} \, dx^i \, dx^j$$
where $( dx^0, dx^1, dx^2, dx^3)$ is a four-dimensional spacetime vector and $g$ is a $4 \times 4$ matrix.

The paper where you found this is discussing a moderately advanced topic in general relativity where the readers are expected to be familiar with the notation. If you are interested in relativity, you should start with an introductory text on special relativity and work towards the more difficult stuff later.

 

[mathman]

I've come across countless equations in which i and j are written as either subscript or superscript or both. I am an AS level student (16 - 17 yrs old), and all I know is that i, j and k are unit vectors, but I've never seen them written as subscript or superscript and I'd like to understand what that notation means and how it differs from what I've seen in class.

This is the equation I am currently trying to decipher:

dl² = g_ij dxⁱ dx^j

Which is in this document:

http://www2.warwick.ac.uk/fac/sci/physics/current/teach/module_home/px436/notes/lecture20.pdf

Thanks in advance ^-^

In this context i and j have nothing to do with the i,j,k unit vectors. They are simply indicies. What you have here is tensor notation, where an index which is repeated implies that you sum over the index. What you have is $dl^2=\sum_{i=1}^3 \sum_{j=1}^3 g_{ij}dx^idx^j$.